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Diffraction Effects

The Airy Disk and the Rayleigh Criterion

As discussed in your lectures, diffraction of light (with a circular aperture) causes point sources of light to appear as disks rather than single points. These disks are known as Airy Disks. The disks appear as a central peak, surrounded by a series of fainter rings, with dark minima in between the rings; the central maximum contains 84% of the total light. Two adjacent sources are said to be just resolved if the central maximum of one source coincides with the first minimum of the other source - this is known as the Rayleigh Criterion. The Rayleigh Criterion therefore defines the minimum separation that two objects can have in order for them to be barely resolved. It is a function of the wavelength of the light and the aperture size of the telescope, and is also known as the angular resolution. It is given by the formula

Rayleigh Criterion equation

Where theta is the (diffraction limited - see below) angular resolution in radians, lambda is the wavelength of the light, and D is the telescope aperture (both have to be in the same units, e.g. metres). The factor of 1.22 results from the aperture being circular.

So if the sources are closer together than this then they are unresolved, and if the sources are further apart then they are resolved (for a specific telescope/wavelength).  For example, the 8-inch (20-cm) Fry telescope, used visually (wavelength about 550 nm), has a resolution limit of about 0.7 arc-seconds by the Rayleigh Criterion.  Double stars closer than this separation will appear as single stars.
See diagram 6-15 in the Universe textbook for diagrams of Airy disks and the Rayleigh Criterion.

Diffraction Limited Optics and Seeing

When building optical instruments, the makers always aim to produce a set of optics that are 'diffraction limited'. By this they mean that the limiting factor on the quality of images produced is the effect of diffraction of light when it passes through the optics - i.e. there are no imperfections in the optics that will produce an effect that significantly detracts from the diffraction resolution. However, even if optics are described as diffraction-limited, this does not mean that the specified angular resolution will be obtainable when observing objects - turbulence in the atmosphere of the earth generally limits the observed angular resolution to approximately 0.25 to 3.0 arc-second - this is known as 'seeing', and the seeing describes the limits the atmosphere places on angular resolution. Obviously this does not affect space-based telescopes since they are above the atmosphere. So for normal use, a telescope with angular resolution better than the seeing will still be limited. There are ways to get around this limit in resolution for ground-based telescopes, such as using Active Optics and/or artificial stars.

The Danjon scale for atmospheric turbulence.

The atmospheric seeing may be characterised using the Danjon scale. The image above illustrates this scale for the case of a single star. The first picture shows an Airy disk for perfect seeing (V). As turbulence increases, the Airy disk breaks down and becomes more chaotic, and poor seeing the image is composed of a number of rapidly moving blobs (I).

Visual planetary observers often use the Antoniadi scale. It also uses Roman numerals like the Danjon scale, but in the reverse direction so that I is the best seeing and V the worst (just to be confusing).

I Perfect seeing, without a quiver.
II Slight undulations, with moments of calm lasting several seconds.
III Moderate seeing, with larger air tremors.
IV Poor seeing, with constant troublesome undulations.
V Very bad seeing, scarcely allowing the makings of a rough sketch.
Source: Norton Star Atlas, 17th Edition. Longman (1986).

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